Question

If $N$ is a set of natural numbers, then under binary operation $a\cdot b=a+b,(N,.)$ is

• A. Quasi-group
• B. Semi-group
• C. Monoid
• D. Group

Answer 1

Answer: (B)

Semi-group

Let $a,b$ be any element in $N$ $⟹$ $a+bϵN$

$\therefore a\cdot b=a+bϵN$

$⟹a\cdot bϵN$ ......... $\left(1\right)$

$(a\cdot b)\cdot c=(a\cdot b)+c=a+b+c$

Also, $a\cdot (b\cdot c)=a+(b\cdot c)=a+b+c$

$\therefore a\cdot (b\cdot c)=(a\cdot b)\cdot c$ ........ $\left(2\right)$

$a\cdot b=a+b=b+a=b\cdot a$ ......... $\left(3\right)$

From $\left(1\right),\left(2\right)$ and $\left(3\right)$,

The structure $(N,.)$ satisfies the closure property, associativity and commutativity but the identity element $0$ does not belong to $N$.

Hence, $(N,.)$ is a semi-group.